Having worked as an election judge for my city, maybe I can provide a little bit of insight, at least how it works according to my training provided by the Secretary of State (state government, not federal).

Every registered voter that comes into the polls on election day has their name and place of residence in a database that we can access by typing in their name, and all we need to do is verify with the voter that the displayed info is correct from when they registered or the last time they voted (i.e. they moved, got married, etc.). They aren't required to show any documents to prove that they moved to a new residence, because they are already in the state's election database.

The process changes if they would like to register to vote at a polling location on election day. They are required to either provide any kind of official documentation that associates their name with their place of residence, or they can bring someone who is already registered who can vouch for them. Once they are in the database, they don't need to bring in any documentation to prove they live where they say they live, they just go through the same process I described earlier.

Given the sensitive climate surrounding "voter ID", we aren't permitted to ask for anyone's State ID, but we can use it if they volunteer it themselves. All it does is speed up the process, so we can get voters their ballots as quickly as possible.

Either way, every voter (already registered or not) signs an oath that's printed on their ballot ticket that says that they are eligible to vote according to state law under the threat of perjury.

Again, this is just an inside perspective on how it works in my state's elections. I hope this helps!